Journal Article

Statistical-Physics-Based Reconstruction in Compressed Sensing

Mézard, M., Sun, Y. F., Zdeborová, L., Sausset, F., Krzakala, F.

2012

Overview

Compressed sensing has triggered a major evolution in signal acquisition. It consists of sampling a sparse signal at low rate and later using computational power for the exact reconstruction of the signal, so that only the necessary information is measured. Current reconstruction techniques are limited, however, to acquisition rates larger than the true density of the signal. We design a new procedure that is able to reconstruct the signal exactly with a number of measurements that approaches the theoretical limit, i.e., the number of nonzero components of the signal, in the limit of large systems. The design is based on the joint use of three essential ingredients: a probabilistic approach to signal reconstruction, a message-passing algorithm adapted from belief propagation, and a careful design of the measurement matrix inspired by the theory of crystal nucleation. The performance of this new algorithm is analyzed by statistical-physics methods. The obtained improvement is confirmed by numerical studies of several cases.

• Publisher: American Physical Society
• Subject: Algorithms / Computed tomography / Computer Science / Condensed Matter / Data acquisition / Data points / Design / Gene sequencing / Image reconstruction / Information Theory / Interdisciplinary aspects / Magnetic resonance / Mapping / Mathematical analysis / Mathematics / Message passing / Nucleation / Optimization / Physics / Pixels / Probability theory / Propagation / Sampling methods / Signal processing / Signal reconstruction / Statistical analysis / Statistical Mechanics / Statistical methods / Statistical physics / Vectors (mathematics)